Matrix discrepancy and the log-rank conjecture

Given an \(m\times n\) binary matrix M with \(|M|=p\cdot mn\) (where |M| denotes the number of 1 entries), define the discrepancy of M as \({{\,\textrm{disc}\,}}(M)=\displaystyle \max \nolimits _{X\subset [m], Y\subset [n]}\big ||M[X\times Y]|-p|X|\cdot |Y|\big |\). Using semidefinite programming and spectral techniques, we prove that if \({{\,\textrm{rank}\,}}(M)\le r\) and \(p\le 1/2\), then

$$\begin{aligned}{{\,\textrm{disc}\,}}(M)\ge \Omega (mn)\cdot \min \left\{ p,\frac{p^{1/2}}{\sqrt{r}}\right\} .\end{aligned}$$

We use this result to obtain a modest improvement of Lovett’s best known upper bound on the log-rank conjecture. We prove that any \(m\times n\) binary matrix M of rank at most r contains an \((m\cdot 2^{-O(\sqrt{r})})\times (n\cdot 2^{-O(\sqrt{r})})\) sized all-1 or all-0 submatrix, which implies that the deterministic communication complexity of any Boolean function of rank r is at most \(O(\sqrt{r})\).